The Generalized Birthday Problem

Saperstein, Bernard

doi:10.1080/01621459.1972.10482403

Cited by 59 publications

(7 citation statements)

References 4 publications

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“…For β > C counting these strings is equivalent to solving the so-called generalized birthday problem. Rather involved closed form expressions forBn(m, C, j) were derived in [14] when j/2 < C and in [13,Theorem 1] for the general case. The latter however are expressed as a large sum of determinants and therefore does not result in an efficient manner to computeBn(m, C, j).…”

Section: It Is Easy To See Thatmentioning

confidence: 99%

Spatial fairness in multi-channel CSMA line networks

Block

Houdt

2016

Performance Evaluation

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In this paper we consider a multi-channel random-access carrier-sense multiple access (CSMA) line network with n saturated links, where each link can be active on at most one of the C available channels at any time. Using the product form solution of such a network, we develop fast algorithms to compute the per-link throughputs and use these to study the spatial fairness in such a network. We consider both standard CSMA networks and CSMA networks with so-called channel repacking.Recently it was shown that fairness in a single channel CSMA line network can be achieved by means of a simple formula for the activation rates, which depends solely on the number of interfering neighbors. In this paper we show that this formula still achieves fairness in the multi-channel setting under heavy and low traffic, but no such simple formula seems to exist in general. On the other hand, numerical experiments show that the fairness index when using the simple single channel formula in the multi-channel setting is very close to one, meaning this simple formula also eliminates most of the spatial unfairness in a multi-channel network.

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Section: It Is Easy To See Thatmentioning

confidence: 99%

Spatial fairness in multi-channel CSMA line networks

Block

Houdt

2016

Performance Evaluation

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“…Table IV contains the best solutions for different values of A * with fixed allocation such that V = (1,2,3,4,5,6,7,8,9,10). The solutions are found by running GA for 1000 iterations with a randomly initialized population.…”

Section: Scenario 1: Fixed Element Allocationmentioning

confidence: 99%

“…This system was formally introduced by Griffith [1]. It was also mentioned in some earlier research [2][3][4][5][6]. Examples of the linear consecutive k-out-of-r-from-n system can be found in quality control, inspection procedures, service systems, and radar detection problems.…”

Section: Introductionmentioning

confidence: 99%

Optimal replacement and allocation of multi‐state elements in k‐within‐m‐from‐r/n sliding window systems

Xiao

Peng

Levitin

2015

Appl Stoch Models Bus & Ind

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This paper proposes a new model that generalizes the linear sliding window system to the case of multiple failures. The considered k-within-m-from-r/n sliding window system consists of n linearly ordered multi-state elements and fails if at least k groups out of m consecutive groups of r consecutive multi-state elements have cumulative performance lower than the demand W. A reliability evaluation algorithm is suggested for the proposed system. In order to increase the system availability, maintenance actions can be performed, and the elements can be optimally allocated. A joint element allocation and maintenance optimization model is formulated with the objective of minimizing the total maintenance cost subjected to the pre-specified system availability requirement. Basic procedures of genetic algorithms are adapted to solve the optimization problem. Numerical experiments are presented to illustrate the applications.

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“…This system was formally introduced by Griffith (1986), but had been mentioned previously by Saperstein (1972Saperstein ( , 1973, Naus (1974), Nelson (1978) and Tong (1985) in connection with tests for non-random clustering, quality control and inspection procedures, service systems, and radar problems. This system was formally introduced by Griffith (1986), but had been mentioned previously by Saperstein (1972Saperstein ( , 1973, Naus (1974), Nelson (1978) and Tong (1985) in connection with tests for non-random clustering, quality control and inspection procedures, service systems, and radar problems.…”

Section: Introductionmentioning

confidence: 99%

Element Availability Importance in Generalizedk-out-of-r-from-nSystems

Levitin¹

2003

IIE Transactions

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An algorithm for evaluating the overall system availability and the availability importance of elements in a generalized consecutive k-out-of-r-from-n:F system is presented. This system consists of n linearly ordered nonidentical elements with different reliability characteristics and performance rates. Each element can have two states: (i) complete failure with a performance rate of zero; and (ii) perfect functioning with a nominal performance rate. The system fails if the sum of the performance rates of any r consecutive elements is lower than a demand W. The suggested algorithm, based on an extended universal moment generating function, is convenient for numerical implementation. Analytical and numerical examples of evaluating availability importance indices are presented.

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The Generalized Birthday Problem

Cited by 59 publications

References 4 publications

Spatial fairness in multi-channel CSMA line networks

Spatial fairness in multi-channel CSMA line networks

Optimal replacement and allocation of multi‐state elements in k‐within‐m‐from‐r/n sliding window systems

Element Availability Importance in Generalizedk-out-of-r-from-nSystems

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